348 Chapter 9. Cooperative transitions in macromolecules[[Student version, January 17, 2003]]
9.7 T 2 Stretching curve of the elastic rod model
Wecan get a useful simplification of the solution to the 3d cooperative chain (see Section 9.4.1′on
page 341) by taking the limit of small link length, → 0 (the elastic rod model).
a. Begin with Equation 9.40. Expand this expression in powers of ,holdingA,f,andwfixed and
keeping the terms of order 1 and 2.
b. Evaluate the estimated eigenvalueλmax,estas a function of the quantityf ̄≡Af /kBT,the
variational parameterw,and other constants, again keeping only leading terms in .Show that
lnλmax,est(w)=const +
A(
−
1
2 w
+coth 2w)(
f ̄−^12 w).The first term is independent offandw,and so won’t contribute to Equation 9.41.
c. Even if you can’t do (b), proceed using the result given there. Use some numerical software
to maximize lnλmax,estoverw,and call the result lnλ∗(f ̄). Evaluate Equation 9.41 and graph
〈z/Ltot〉as a function off.Also plot the high-precision result (Equation 9.42) and compare to your
answer, which used the Ritz variational approximation.
9.8 T 2 Low-force limit of the elastic rod model
a. If you didn’t do Problem 9.7, take as given the result in (b). Consider only the case of very low
applied force,f kBT/A.Inthis case you can do the maximization analytically (on paper). Do
it, find the relative extension using Equation 9.41, and explain why you “had to” get a result of
the form you did get.
b. In particular, confirm the identificationLseg=2Aalready found in Section 9.1.3′on page 338
bycomparing the low-force extension of the fluctuating elastic rod to that of the 3d freely jointed
chain (see Your Turn 9o).
9.9 T 2 Twist and pop
Astretched macroscopic spring pulls back with a forcef,which increases linearly with the extension
zasf=−kz.Another familiar example is thetorsional spring:It resiststwistingbyan angleθ,
generating atorque
τ=−ktθ. (9.46)
Herektis the “torsional spring constant.” To make sure you understand this formula, show that
the dimensions ofktare the same as those of energy.
It is possible to subject DNA to torsional stress too. One way to accomplish this is using an
enzyme calledligase,which joins the two ends of a piece of DNA together. The two sugar-phosphate
backbones of the DNA duplex then form two separate, closed loops. Each of these loops can bend
(DNA is flexible), but they cannot break or pass through each other. Thus their degree of linking
is fixed—it’s a “topological invariant.”
If we ligate a collection of identical, open DNA molecules at low concentration, the result is a
mixture of various loop types (topoisomers), all chemically identical but topologically distinct.^13
Each topoisomer is characterized by alinking numberM.IfwemeasureMrelative to the most
relaxed possibility, then we can think of it as the number of extra turns that the DNA molecule had
at the moment when it got ligated.Mmay be positive or negative; the corresponding total excess
angle isθ=2πM.Wecan separate different topoisomers using electrophoresis: For example, a
figure-8 is more compact, and hence will migrate more rapidly, than a circular loop.
(^13) Athigher concentration we may also get some double-length loops.